A balancing property of infinite subsets of $\mathbb{N}$

Let $$\omega$$ denote the set of non-negative integers and let $$[\omega]^\omega$$ be the collection of infinite subsets of $$\omega$$.

If $$S\in [\omega]^\omega$$ and $$A\subseteq \omega$$ we say that $$A$$ is well-balanced with respect to $$S$$ if $$\lim_{n\to\infty}\frac{|A\cap S\cap \{1,\ldots,n\}|}{|S\cap\{1,\ldots,n\}|+1} = 1/2.$$ Intuitively speaking, this means that $$A$$ contains "half of the members of $$S$$" (which also implies that $$A$$ is infinite.)

Question. Given $${\frak S}\subseteq [\omega]^\omega$$ with $$|{\frak S}| = \aleph_0$$, is there $$A\in[\omega]^\omega$$ such that $$A$$ is well-balanced with respect to every member of $${\frak S}$$?

• If we take ${\frak S}$ to be the collection of infinite recursive sets of $\omega$, then any such $A$ would correspond to a bitstream that is computationally random. – Dominic van der Zypen Jan 7 at 10:42
• $\liminf a_n=\limsup a_n=1/2$ can be abbreviated as $\lim a_n=1/2$. – GH from MO Jan 7 at 10:45
• Oh right :-) Will simplify the post accordingly, thanks @GHfromMO – Dominic van der Zypen Jan 7 at 10:46
• A random subset of $\omega$ (meaning that $n\in A$ iff the $n^{\text{th}}$ toss in an infinite sequence of fair coin tosses comes of heads) will do the job with probability $1$. – bof Jan 7 at 10:56
• Isn't this sometimes called relative density (of the set $A$ with respect to the set $S$)? – Martin Sleziak Jan 7 at 10:57

By the strong law of large numbers, if $$S$$ is an infinite subset of $$\omega$$, a random subset of $$\omega$$ will be well-balanced with respect to $$S$$ with probability one.
By the countable additivity of Lebesgue measure, if $$\mathfrak S$$ is a countable collection of infinite subsets of $$\omega$$, a random subset of $$\omega$$ will be well-balanced with respect to every member of $$\mathfrak S$$ with probability one.
If Lebesgue measure is $$\kappa$$-additive (the union of fewer than $$\kappa$$ measure zero sets has measure zero) then the same holds for a collection $$\mathfrak S$$ of infinite subsets of $$\omega$$ with $$|\mathfrak S|\lt\kappa$$.